Image Processing Reference
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with their passbands (stopbands) centred at one point of an equidistant frequency grid
f n
8 ,
=
·
=
f c
c
c
0, 1, 2, 3, 4, 5, 6, 7.
(1)
In addition, it is shown that the complex HBF defined by (1) require roughly the same amount
of computation as their original real HBF prototype ( f c
=
f 0 =
0). Especially, we present the
most efficient elementary SFG for sample rate alteration, their main application. The SFG will
be given for LP FIR [Göckler (1996b)] as well as for MP IIR HBF for real- and complex-valued
input and/or output signals, respectively. Detailed comparison of expenditure is included.
In Section 3 we combine two of those linear-phase FIR HBF investigated in Section 2
with different centre frequencies out of the set given by (2), to construct efficient SFG
of directional filters (DF) for separation of one input signal into two output signals
or for combination of two input signals to one output signal, respectively. These DF
are generally referred to as two-channel frequency demultiplexer (FDMUX) or frequency
multiplexer (FMUX) filter bank [Göckler & Eyssele (1992); Vaidyanathan & Nguyen (1987);
Valenzuela & Constantinides (1983)].
In Section 4 of this chapter we consider the application of the two-channel DF as a
building block of a multiple channel tree-structured FDMUX filter bank according to Fig.
2, typically applied for on-board processing in satellite communications [Danesfahani et al.
(1994); Göckler & Felbecker (2001); Göckler & Groth (2004); Göckler & Eyssele (1992)]. In case
of a great number of channels and/or challenging bandwidth requirements, implementation
of the front-end DF is crucial, which must be operated at (extremely) high sampling rates. To
cope with this issue, in Section 4 we present an approach to parallelise at least the front end of
the FDMUX filter bank according to Fig. 2.
2. Single halfband filters 1
In this Section 2 of this chapter we recall the properties of the well-known HBF with real
coefficients (real HBF with centre frequencies f c
according to (1)), and
investigate those of the complex HBF with their passbands (stopbands) centred at
∈{
f 0 , f 4
} = {
0, f n /2
}
f n
8 ,
=
·
=
f c
c
c
1, 2, 3, 5, 6, 7
(2)
f 0 =
0). In particular, we derive the most efficient elementary SFG for sample rate alteration. These
will be given both for LP FIR [Göckler (1996b)] and MP IIR HBF for real- and complex-valued
input and/or output signals, respectively. The expenditure of all eight versions of HBF
according to (1) is determined and thoroughly compared with each other.
The organisation of Section 2 is as follows: First, we recall the properties of both classes of
the afore-mentioned real HBF, the linear-phase (LP) FIR and the minimum-phase (MP) IIR
approaches. The efficient multirate implementations presented are based on the polyphase
decomposition of the filter transfer functions [Bellanger (1989); Göckler & Groth (2004); Mitra
(1998); Vaidyanathan (1993)]. Next, we present the corresponding results on complex HBF
(CHBF), the classical HT, by shifting a real HBF to a centre frequency according to (2) with
c
=
that require roughly the same amount of computation as their real HBF prototype ( f c
. Finally, complex offset HBF (COHBF) are derived by applying frequency shifts
according to (2) with c
∈ {
2,6
}
, and their properties are investigated. Illustrative design
examples and implementations thereof are given.
∈ {
1,3,5,7
}
1 Underlying original publication: Göckler & Damjanovic (2006b)
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